Mathematical Logic, Interpretation, Satisfiable, Consequence relation

Every interpretation which is a model of A is also a model of a iff
not (Sat A) U {~a}

Proof

Every interpretation which is a model of A is also a model of a
iff(1) there is no interpretation which is a model of A but not a model of a
iff(2) there is no interpretation which is model of A U {~a}
iff(3) not (Sat A) U {~a}

i dont understand on the iff(2), it seems to assume that "but" is "union"(U)

this is from ebbinghaus mathematical logic, this is my first time reading mathematical logic books, i've been noticing that much of the proof are using argument(english) rather than formal language. Furthermore like for example iff(1). i can tell how => goes, but rather now i am still trying to accept the <= part. HELP !!
Edit: ok i already get the <= part in iff(1), but still dont know what happen on iff(2)
Edit(2):
the statement "Every interpretation which is a model of A is also a model of a"

let B be arbitrary interpretation

is the statement equivalent to "B is a model of A iff B is a model of a"?
or "B is a model of A => B is a model of a"?